Problem: You have found the following ages (in years) of all 5 sloths at your local zoo: $ 5,\enspace 3,\enspace 5,\enspace 1,\enspace 4$ What is the average age of the sloths at your zoo? What is the variance? You may round your answers to the nearest tenth.
Because we have data for all 5 sloths at the zoo, we are able to calculate the population mean $({\mu})$ and population variance $({\sigma^2})$ To find the population mean , add up the values of all $5$ ages and divide by $5$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{5}} x_i}{{5}} $ $ {\mu} = \dfrac{5 + 3 + 5 + 1 + 4}{{5}} = {3.6\text{ years old}} $ Find the squared deviations from the mean for each sloth. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $5$ years $1.4$ years $1.96$ years $^2$ $3$ years $-0.6$ years $0.36$ years $^2$ $5$ years $1.4$ years $1.96$ years $^2$ $1$ year $-2.6$ years $6.76$ years $^2$ $4$ years $0.4$ years $0.16$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{1.96} + {0.36} + {1.96} + {6.76} + {0.16}} {{5}} $ $ {\sigma^2} = \dfrac{{11.2}}{{5}} = {2.24\text{ years}^2} $ The average sloth at the zoo is 3.6 years old. The population variance is 2.24 years $^2$.